Question

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Answer 1

Step-by-step explanation:

Given that

X=-2 is a root so

3(-2)^2 - 14+p=0

12-14+p=0

P=2

So the equation is

3x^2+7x+2=0

Now we need to find k

We know that sum of roots fornula is - b/a

For first equation it is - 7/3

And second equation it is - 4k

Also given that roots are equal for two equations

So 7/3 =4k

K=7/28

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Answer 2

Given :-----

• x = (-2) is a factor of 3x²+7x+p = 0

To Find :----

• Value of k for which roots are Equal ..

Concept used :--------

→ if x = a is the roots of Equation p(x) , than , p(a) will be Equal to zero ...

→ If A•x^2 + B•x + C = 0 ,is any quadratic equation,

then its discriminant is given by;

D = B^2 - 4•A•C

• If D = 0 , then the given quadratic equation has real and equal roots.

• If D > 0 , then the given quadratic equation has real and distinct roots.

• If D < 0 , then the given quadratic equation has unreal (imaginary) roots.

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Solution :----

since x=-2 is factor of 3x²+7x+p=0 , hence , f(-2) = 0

→ f(-2) = 3(-2)² + 7*(-2) + p = 0

→ f(-2) = 3*4 -14 + p = 0

→ (-2) + p = 0

→ p = 2 ...

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Now, root of Equation x²+k(4x+k-1) + p = 0 are Equal .

Hence,

D = b² - 4ac = 0

Here,

x² + 4kx + (k² - k + p) = 0

→ a = 1

→ b = 4k

→ c = (k² - k + p) ,putting p = 2 now , (k²-k+2)

Putting values now we get,

⇒ 16k² – 4*1*(k² – k + 2) = 0

⇒ 4(4k² - k² +k -2) = 0

⇒ 3k² + k – 2 = 0

⇒ 3k² +3k - 2k -2 = 0

⇒ 3k(k+1)-2(k+1) = 0

⇒ (3k – 2)(k + 1) = 0

Putting both Equal to zero now, we get,

➠ 3k - 2 = 0 ,

➠ k = 2/3 ....

and,

➠ k + 1 = 0

➠ k = (-1)

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Hence, value of k will be (2/3) and (-1) ..

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